Finite Group Databasebeta

← #31796191

Subgroups — #31796191

Order 17536.

242 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578425C211yesmaximal, characteristic, complemented
2C4×C4⋊#10587739C211yesmaximal, characteristic, complemented
3#22578256C212yesmaximal, complemented
4#22578254C212yesmaximal
5#22577935C211yesmaximal, characteristic
6C2×C4⋊#10587739—21no—
7C2×C4×#10587739—21no—
8C4⋊#11764047—21no—
9#17984145C2212yes—
10#17984143—21no—
11(C2×#10587739)⋊C4C2212yes—
12#17984081—21no—
13#17984079C411yescharacteristic, complemented
14#17984075C411yescharacteristic, complemented
15#17984070—21no—
16#17984068C2211yescharacteristic
17#17984059C2211yescharacteristic
18C4×Dic274C411yescharacteristic, complemented
19C4×C22⋊D137—21no—
20#17984047—21no—
21C42×D137C2211yescharacteristic
22C4×D4×C137C411yescharacteristic, complemented
23C22×#10587739—22no—
24C2×#11764047—22no—
25C4⋊#10587739—24no—
26C4×#10587739—42no—
27C2×Dic274C2×C411yescharacteristic
28C2×C22⋊D137—21no—
29C2×C4×D137D411yescharacteristic
30C2×C4×D137D411yescharacteristic
31C2×C4×D137C2311yescharacteristic
32C4⋊Dic137—21no—
33C4⋊Dic137—21no—
34C4⋊Dic137C2×C411yescharacteristic
35#14555489—21no—
36#14555489—21no—
37C22⋊Dic137—21no—
38C22⋊Dic137—21no—
39C22⋊Dic137—21no—
40#14555487—21no—
41C4×Dic137—21no—
42C4×Dic137C2×C411yescharacteristic
43C4×Dic137D411yescharacteristic
44C4×Dic137D411yescharacteristic
45C2×D4×C137C2×C411yescharacteristic
46C22×C4×C137—21no—
47C4⋊C4×C137C2×C411yescharacteristic
48C22⋊C4×C137—21no—
49C42×C137C2×C411yescharacteristic
50C2×#10587739—24no—
51C2×#10587739—42no—
52C2×#10587739—42no—
53#11764047—42no—
54C22×D137C2×D411yescharacteristic
55Dic274—41no—
56C22⋊D137—41no—
57C22⋊D137—41no—
58C2×Dic137—21no—
59C2×Dic137—21no—
60C2×Dic137C22⋊C411yescharacteristic
61C2×Dic137C22⋊C411yescharacteristic
62C2×Dic137C2×D411yescharacteristic
63C4×D137—21no—
64C4×D137—21no—
65C4×D137—22no—
66C4×D137—41no—
67C23×C137—21no—
68D4×C137—41no—
69C2×C4×C137—21no—
70C2×C4×C137—42no—
71C2×C4×C137C22⋊C411yescharacteristic
72C2×C4×C137C22⋊C411yescharacteristic
73C2×C4×C137C22×C411yescharacteristic
74C22×C137—41no—
75C22×C137—41no—
76C22×C137—41no—
77C22×C137C2×C22⋊C411yescharacteristic
78C2×D137—21no—
79C2×D137—21no—
80C2×D137AGL(1,ℤ/8)11yescharacteristic
81C2×D137#13711yescharacteristic
82#10587739—44no—
83C4×C137—21no—
84C4×C137—21no—
85C4×C137—41no—
86C4×C137#10412yes—
87Dic137—21no—
88Dic137—21no—
89Dic137—21no—
90Dic137—41no—
91Dic137—41no—
92C2×C137—81no—
93C2×C137#41011yescharacteristic
94C2×C137#41711yescharacteristic
95C2×C137C2×#10411yescharacteristic
96D137—21no—
97D137—21no—
98C137#150911yescharacteristic, complemented, Sylow
99#1509—1371nomaximal, Sylow
100#529—1371no—
101#432—1371no—
102C4×C4⋊C4—1371no—
103#328—1372no—
104#327—1372no—
105#126—2741no—
106#125—1371no—
107#124—1371no—
108#123—2741no—
109C22⋊Q8—1371no—
110#121—1371no—
111C4×Q8—1371no—
112C4×D4—2741no—
113C4×D4#1058773911yescharacteristic, complemented
114#117—2741no—
115C2×C4⋊C4—2741no—
116C2×C42—2741no—
117C2×C42—1371no—
118C4⋊C8—2741no—
119C22⋊C8—1372no—
120#97—2741no—
121#95—1372no—
122C2×Q8—1371no—
123C2×D4—2741no—
124C2×D4C2×#1058773911yescharacteristic
125C22×C4—21no—
126C22×C4—2742no—
127C22×C4—1371no—
128C22×C4—1371no—
129C22×C4—1371no—
130C2×C8—2742no—
131C4⋊C4—2741no—
132C4⋊C4—2741no—
133C4⋊C4—2741no—
134C4⋊C4—2744no—
135C4⋊C4—2741no—
136C4⋊C4—1371no—
137C4⋊C4C2×#1058773911yescharacteristic
138C22⋊C4—21no—
139C22⋊C4—2741no—
140C22⋊C4—2741no—
141C22⋊C4—2741no—
142C22⋊C4—2741no—
143C42—2741no—
144C42—5482no—
145C42—1371no—
146C42—1371no—
147C42—1371no—
148C42C2×#1058773911yescharacteristic
149C23—21no—
150C23—1371no—
151Q8—5481no—
152D4—41no—
153D4—5481no—
154D4—5481no—
155C2×C4—21no—
156C2×C4—42no—
157C2×C4—2741no—
158C2×C4—2741no—
159C2×C4—2742no—
160C2×C4—2744no—
161C2×C4—2741no—
162C2×C4—2741no—
163C2×C4—5481no—
164C2×C4—5482no—
165C2×C4—5482no—
166C2×C4—1371no—
167C2×C4—1371no—
168C2×C4—1371no—
169C2×C4C22⋊#1058773911yescharacteristic
170C2×C4C22⋊#1058773911yescharacteristic
171C2×C4C22×#1058773911yescharacteristic
172C8—5482no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—2741no—
177C22—2741no—
178C22—1371no—
179C22—1371no—
180C22C2×C22⋊#1058773911yescharacteristic
181C4—21no—
182C4—21no—
183C4—41no—
184C4—2741no—
185C4—2741no—
186C4—2741no—
187C4—5484no—
188C4—5481no—
189C4—5481no—
190C4#1798415212yes—
191C2—81no—
192C2—2741no—
193C2—2741no—
194C2#2257838211yescharacteristic
195C2#2257839411yescharacteristic
196C2C2×#1798415211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4×D4×C137C4
socleC22×C137C2×C22⋊C4
prime radical#17984059C22
cosocleC2×C4×D137C23
hypercenterC4×D4#10587739
hypocenterC137#1509
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22578425C2semidirect
2. C4×C4⋊#10587739C2semidirect
3. #22578256C2semidirect
13. #17984079C4semidirect
14. #17984075C4semidirect
18. C4×Dic274C4semidirect
22. C4×D4×C137C4semidirect
98. C137#1509semidirect
113. C4×D4#10587739semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225784252
C4×C4⋊#105877392
#225782562
#225782542
#225779352
#1509137